In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic, and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot, and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer.
When ready to use, the cooked pasta should be added to a saucepan with the appropriate
amount of sauce. Garnish with basil leaves and cheese, if using.